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Asteroid Gaspra as imaged by Galileo during a close fly-by. Gaspra is about 12x20 km.

Best Observing Season: any Level: Intermediate to Advanced Learning Goals: The student will learn to find the color, type and size of an asteroid. Terminology: abundances, albedo, C-type asteroid, M-type asteroid, opposition, phase angle, prograde, retrograde, S-type asteroid Software: Megastar, MaxIm Archive Image Directory: asteroid/props Archive Image List: B and V images of C and S-type asteroids, two images separated by one to three hours. References: Kowal, C. Asteroids; The Astronomical Almanac

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Background and Theory

Asteroids are relatively small chunks of rock orbiting the sun mostly between the orbits of Mars and Jupiter.  They are also known as minor planets. The first asteroid was discovered by Giuseppe Piazzi of Palermo, Sicily in 1801 and was later named Ceres (Greek goddess of agriculture and patron of Sicily).  Ceres is the largest asteroid with a diameter of 950 km, about 30% the diameter of the Moon.  More than 9,000 asteroids are catalogued today, most of them much smaller than Ceres.

The albedo of an asteroid is the fraction of incident light which is reflected by the object.  An object with an albedo smaller than 0.1 (10%) would look like a piece of black coal, while one with an albedo of 50% would look like white chalk. There are three classes of asteroids: S-type which appear light (albedos 10-20%) and have color index 0.8<B-V<1.0. They are mostly located near the orbit of Mars. C-type asteroids are quite black (albedos 2-6%) with color indices 0.6<B-V<0.8. The M-type (albedos 10-18%, 0.7<B-V<0.8) have high metal abundances. About 75% of all asteroids are S-type.

The shape of the largest asteroids is nearly spherical, but most asteroids are small (less than 100 km diameter) and irregularly shaped. An image of the asteroid Gaspra (about 10x15 km) taken by the spacecraft Galileo is shown in figure 1. Most asteroids rotate with periods of 5-50 hours, so the apparent brightness can change rapidly. The apparent magnitude will also depend on distance,  albedo, and phase angle as described below. The brightest asteroids have apparent magnitudes V~6-8 near opposition [1] , but most are considerably fainter. A good summary of asteroid properties and history can be found in the book Asteroids by Charles Kowal.

Procedure

Observing

1.     Choose a bright (V<11) asteroid which transits within a couple of hours of local midnight for the date and time of observation. (This ensures that the asteroid is near opposition). This can be done mostly conveniently using the sky display program Megastar, which has over 5,500 asteroids available.

2.     Consult the Astronomical Almanac’s list of UBVRI standard stars to find a standard star near your asteroid on the date of observation.  Request images of this star in B, V, R and I filters.  You will probably find it easiest to manually enter the coordinates of the standard star.  Make a note of the R.A. and Dec of the chosen standard star, as well as its magnitude in each of the 4 filters.  Be careful about your minus signs when translating color indices (B-V) into magnitudes (B). 

3.     Request images of the asteroid you have chosen.. The asteroid name must be given as a number (e.g. 3  for Juno or 725 for Amanda).  Specify asteroid as the catalog file. Take two sets of images using C, B, V, R and I filters separated by at least one hour. Make sure you are careful with the exposure times - use the apparent magnitude given by Megastar as a guide. 

Image Analysis

Part A: Measuring Angular Motion

In this section you will measure the angular motion of the asteroid between two (or more) images and find the approximate distance to the asteroid.

1.      Check the telescope log to determine the exact celestial coordinates, and date and time of observation of the first clear filter image. Make a printed finding chart centered on those coordinates using Megastar. Use the grid option to draw gridlines. Mark the expected position of the asteroid.

2.      Align the C filter images so they have the same pixel coordinates. To align the images, use the Align tool described in the MaxIm portion of Appendix C.

3.      Measure the angular motion of the asteroid in arcseconds per second.  First record the coordinates of the asteroid in the first image.  Then click on the asteroid in the second image and record its coordinates.  Use the Pythagorean Theorem:

 to get the distance between the two points, then check the headers for the times the images were taken.  Divide the distance by the time (in sec) and you have a velocity in pixels per sec.  Multiply by the image scale to get the velocity in arcseconds per second.  Multiply by the number of seconds in an hour. This yields the angular velocity in arcseconds per hour.

4.      The relationship between the orbital period of any smallish solar system object in a circular orbit [2] and the radius of the orbit can be written using Kepler’s third law as

5.      For an asteroid near opposition in a prograde orbit, the observed angular motion as viewed from the Earth is the difference between the angular motion of the Earth and the object’s motion. The motion is retrograde (moves to the right on image - why?).  The relationship between the asteroid’s distance from the Sun, d, and the observed angular motion at the Earth can be calculated using the law of sines and some algebra. 

where

 

and the measured angular motion w  is measured in arcseconds per hour.

6.      Use this formula to estimate the Sun-asteroid distance. Compare with the distance as determined by Megastar  (press F8 key, enter asteroid number and the observation date and time in UT - from FITS header). Note that if the asteroid is far from opposition or has a large eccentricity, this estimate will not be very accurate.

Part B: Determining the Color and Type

1.    Use differential photometry to determine the B and V magnitudes of the asteroid.  To do this:

a)                  Examine your finding chart from Megastar and the images of your asteroid and standard star.  Locate the standard star on the image of your asteroid by comparing the fields. 

b)                  Load the B filter image of your asteroid.

c)                  Select View/Information Window from the menu.  Click on Calibrate.  Make sure your target circle is large enough to enclose the entire asteroid by adjusting the size with the right mouse button.

d)                  Click on your standard star and enter its B magnitude in the box provided.  Click OK.

e)                  Move your cursor over the asteroid.  Record its magnitude.

f)                    Repeat the above steps for the V filter image.

2.    Calculate the color index B-V. Using the information from the Background and Theory section, estimate the asteroid type.

Part C: Estimating the Diameter

The apparent magnitude of an asteroid depends on several factors.  An approximate formula for estimating the peak apparent visual magnitude near opposition is [3]

where V is the apparent visual magnitude near opposition, s is the asteroid radius in km,  r and , in a.u., are the  distances from the asteroid to the Sun and Earth respectively, and A is the albedo. Asteroid albedos vary from nearly zero to over 0.40.  Solving for s, the radius of the asteroid, gives:

            For observations away from opposition, the observed brightness will be fainter than that given above because only a fraction of the surface visible from Earth is lit by the Sun (that is the asteroid is not at “full” phase). We can compute [4] the phase angle

            The reduction in apparent magnitude depends on the type of asteroid one is observing, but a reasonably approximate correction can be written as

where   is measured in degrees. [5]

1.      Calculate the phase angle as described above using the distances from Megastar.

2.      Calculate the expected apparent magnitude at opposition V. Compare with your measured V.

3.      Use your estimate of the asteroid type and the discussion in the introduction to estimate the albedo.

4.       Calculate the diameter of your asteroid by solving for s in the first equation. Compare with the value given in Kowal’s appendix.